Diameter and Stationary Distribution of Random $r$-Out Digraphs

Let $D(n,r)$ be a random $r$-out regular directed multigraph on the set of vertices $\{1,\ldots,n\}$. In this work, we establish that for every $r \ge 2$, there exists $\eta_r>0$ such that $\mathrm{diam}(D(n,r))=(1+\eta_r+o(1))\log_r{n}$. The constant $\eta_r$ is related to branching processes and also appears in other models of random undirected graphs. Our techniques also allow us to bound some extremal quantities related to the stationary distribution of a simple random walk on $D(n,r)$. In particular, we determine the asymptotic behaviour of $\pi_{\max}$ and $\pi_{\min}$, the maximum and the minimum values of the stationary distribution. We show that with high probability $\pi_{\max} = n^{-1+o(1)}$ and $\pi_{\min}=n^{-(1+\eta_r)+o(1)}$. Our proof shows that the vertices with $\pi(v)$ near to $\pi_{\min}$ lie at the top of "narrow, slippery tower"; such vertices are also responsible for increasing the diameter from $(1+o(1))\log_r n$ to $(1+\eta_r+o(1))\log_r{n}$.